Chapter 2 Solutions

These worked solutions are authored for the reviewed prompts shown here. Any added geometry or assumption is stated. Original chapter problems supplied no answer key, so these solutions are separate from the archived manuscript.

Original problem 1 · worked solution

Four point charges each equal to −Q, Q>0, are at (±a,±a), with a=1 m. Find the potential at the origin using V(∞)=0.

Authored clarification: The original says “negative charges of charge q”; Q here is the positive magnitude, so each source is −Q.

All source distances are equal.
$r=\sqrt{a^2+a^2}=\sqrt2a$ .
Potential adds as a scalar.
$V(0)=4{k(-Q)\over\sqrt2a}=-{2\sqrt2kQ\over a}$ .

Answer

$V(0)=-2\sqrt2kQ/a$ .

Check

Potential is negative for four negative sources, despite the electric field at the symmetric center being zero.

Practice prompt · Original reading location

Archived original prompt

Four negative charges of charge q are placed at the positions (-1,-1), (-1, 1), (1,-1), and (1,1). Find the electric potential at the point (0,0).

Original problem 2 · worked solution

Optional original-source extension. Prerequisites: ideal infinite-sheet fields, signed potential differences and energy conservation. This item makes no AP coverage claim.

Infinite sheets at x=0 and x=d>0 have surface densities +σ and +2σ. Find V(d)−V(0) and the energy change for a test charge q moved from 0 to d. Which charge sign could accelerate from rest in that direction?

Authored clarification: σ>0 denotes surface density, not a finite total charge. The motion starts at the +σ sheet and ends at the +2σ sheet.

Superpose the sheet fields between them.
$\vec E={\sigma\over2\epsilon_0}\hat x-{2\sigma\over2\epsilon_0}\hat x=-{\sigma\over2\epsilon_0}\hat x$ .
Integrate the potential difference.
$V(d)-V(0)=-\int_0^dE_x\,dx={\sigma d\over2\epsilon_0}$ .
Relate energy and charge sign.
$\Delta U=q\Delta V={q\sigma d\over2\epsilon_0}$ . A negative q has force in +x and loses potential energy.

Answer

$\Delta V=\sigma d/(2\epsilon_0),\quad\Delta U=q\sigma d/(2\epsilon_0)$ ; a negative charge can accelerate toward the +2σ sheet.

Check

A positive charge would need external energy for quasistatic motion in this direction.

Practice prompt · Original reading location

Archived original prompt

Two infinite plates are placed near one another. One plate has a charge of 2σ the other has a charge of σ. What is the energy associated with moving a particle with charge q from the plate with charge +σ to the plate with charge +2σ? What type of charge would the particle need to move between the plate without outside help?

Original problem 3 · worked solution

Optional original-source extension. Prerequisites: scalar potential superposition for ideal thin rings and a reference at infinity. This item makes no AP coverage claim.

Two thin uniformly charged rings of radius R share a center and lie in perpendicular planes. Their total charges are Q₁ and Q₂. Find the potential at the common center with V(∞)=0.

Authored clarification: The original does not give the ring charges; Q₁ and Q₂ are explicit independent parameters.

Every element of each ring is distance R from the center.
$V_i=k\int dq/R=kQ_i/R$ .
Add scalar potentials.
$V={k(Q_1+Q_2)\over R}$ .

Answer

$V=k(Q_1+Q_2)/R$ . Perpendicular orientation does not alter the equal element distances.

Check

Equal and opposite total charges give zero potential; equal charge Q on both gives 2kQ/R.

Practice prompt · Original reading location

Archived original prompt

Two charged ring of radius z are placed so that they are perpendicular and their centers coincide. What is the electric potential at the center of this setup?

Original problem 4 · worked solution

Optional original-source extension. Prerequisites: continuous surface charge, a definite integral and electric potential energy. This item makes no AP coverage claim.

A thin disk of radius R and uniform total charge Q lies in the xy plane. Find V(h) on its axis for h≥0 with V(∞)=0, and ΔU for a test charge qₜ moved quasistatically from the center just above the disk to height h.

Authored clarification: The disk charge Q and moving test charge qₜ are distinct. The initial location and potential reference are now specified.

Build the disk from concentric rings.
$\sigma=Q/(\pi R^2),\quad dq=2\pi\sigma s\,ds$ .
Integrate potential, not its vector components.
$V(h)=2\pi k\sigma\int_0^R{s\,ds\over\sqrt{s^2+h^2}}={\sigma\over2\epsilon_0}(\sqrt{R^2+h^2}-h)$ .
Subtract the initial potential and multiply by test charge.
$\Delta U={q_t\sigma\over2\epsilon_0}(\sqrt{R^2+h^2}-h-R)$ .
Expand for h≫R.
$\sqrt{h^2+R^2}-h\simeq R^2/(2h)\quad\Rightarrow\quad V(h)\simeq kQ/h$ .

Answer

V(h) and ΔU are the displayed expressions; quasistatic external work equals ΔU.

Check

For positive Q and qₜ, moving away decreases potential energy. V(∞)=0 is valid because the disk is finite.

Practice prompt · Original reading location

Archived original prompt

A disk of radius z with a uniformly distributed charge q sits on the ground. What is the electric potential and electric potential energy associated with moving a charge a height h above its center? What do you expect to see when h>>z?

Original problem 5 · worked solution

Optional original-source extension. Prerequisites: an ideal infinite-line field, a radial potential integral and logarithms. This item makes no AP coverage claim.

An infinitely long wire has uniform line charge density λ. Determine V(d) with the finite reference V(r₀)=0.

Authored clarification: The reference radius r₀>0 and line density λ are essential; an ideal infinite line does not admit V(∞)=0.

Use cylindrical symmetry and Gauss’s law.
$E_r=\lambda/(2\pi\epsilon_0r)$ .
Integrate from the stated reference radius.
$V(d)-V(r_0)=-\int_{r_0}^d{\lambda\over2\pi\epsilon_0r}\,dr={\lambda\over2\pi\epsilon_0}\ln(r_0/d)$ .

Answer

$V(d)={\lambda\over2\pi\epsilon_0}\ln(r_0/d)$ .

Check

V(r₀)=0 and −dV/dd reproduces Eᵣ. The logarithm’s argument is dimensionless.

Practice prompt · Original reading location

Archived original prompt

Find the potential a distance d away from a charged infinitely long wire.

Original problem 6 · worked solution

Define an equipotential and describe the equipotential surfaces of an isolated point charge Q using V(∞)=0.

Authored clarification: Treat Q as a nonzero isolated point charge in three dimensions. Describe the full surfaces and a cross section in a plane through the charge.

Use the defining property.
An equipotential is the set of points with the same scalar potential V.
Solve the point-charge relation.
$V=kQ/r$ is constant at fixed r, so each nonzero admissible value selects a sphere centered on the charge.
Relate to work and field.
Motion along the surface has ΔV=0, hence electric work is zero. The field is normal to the surface because $\vec E=-\nabla V$ .

Answer

Draw concentric circles in a plane through the charge, labeling their potentials kQ/r; they represent concentric spheres.

Check

Equal spacing in V generally gives unequal spacing in r.

Source wording and interpretation: A full three-dimensional surface is a sphere; a planar drawing shows circular cross sections.

Practice prompt · Original reading location

Archived original prompt

Define what an equipotential is and then the draw equipotentials for a point charge.

Original problem 7 · worked solution

Explain the relation between equipotentials and electric field lines. For a negative point charge −Q, Q>0, does potential increase or decrease as radius increases?

Authored clarification: The source sentence ends unfinished. This explicitly completes the comparison using increasing radius and V(∞)=0.

A tangential derivative on an equipotential is zero.
$\vec E=-\nabla V$ is perpendicular to the equipotential where the field is nonzero.
Differentiate the negative-charge potential.
$V=-kQ/r,\qquad dV/dr=kQ/r^2>0$ .
Connect direction and potential change.
Potential increases toward zero as radius increases. The electric field points inward, toward decreasing potential.

Answer

Equipotentials are orthogonal to field lines; the potential of a negative point charge increases with increasing radius.

Check

This is consistent with Eᵣ=−kQ/r² and V approaching zero from below.

Practice prompt · Original reading location

Archived original prompt

Explain how the equipotential relates to electric field lines. Do the equipotentials of a negative charge increase of

Original problem 8 · worked solution

Optional original-source extension. Prerequisites: partial derivatives and the potential-gradient relation. This item makes no AP coverage claim.

A potential is $V(x,y,z)=a x^4y^3+by^2+c$ . Find the electric field.

Authored clarification: This transcribes the original formula image with coefficients a and b to make its units explicit; the source image corresponds to a=b=1 in its chosen units.

Apply the negative gradient.
$\vec E=-\nabla V$ .
Take each partial derivative while holding other coordinates fixed.
$\partial_xV=4ax^3y^3,\quad\partial_yV=3ax^4y^2+2by,\quad\partial_zV=0$ .

Answer

$\vec E=-4ax^3y^3\hat x-(3ax^4y^2+2by)\hat y$ .

Check

The constant c does not affect the field, and the minus sign applies to both nonzero derivatives.

Practice prompt · Original reading location

Archived original prompt

The potential of some electric field is defined by the equation

Find the electric field associated with this potential.

Original problem 1 · worked solution

Original problem 2 · worked solution

Original problem 3 · worked solution

Original problem 4 · worked solution

Original problem 5 · worked solution

Original problem 6 · worked solution

Original problem 7 · worked solution

Original problem 8 · worked solution

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